On the probability of generating prosoluble groups

A profinite group G is said to be 'positively finitely generated', PFG for short, if the probability that k random elements generate G is positive for some natural number k, and the least such natural number is denoted by dp (G). Another invariant appears naturally in this context. A profinite group is PFG if and only if it has polynomial maximal subgroup growth; for such a group G there is a natural definition, given by A. Mann, of the degree s(G) of maximal subgroup growth. We prove the following: Let F be a free prosoluble group of rank d with d > 2; then c3d - c3 + 1 < s(F) <_ dp (F)<_ max{c3d - c3 + 1, c2d}, where c2 and c3 are explicitly known constants.